Quantization of symplectic manifolds with conical points

Quantization of a general nonlinear phase manifold $\mathfrak X$ in the quasicIassical approximation leads to the two-dimensional analog of the Bohr–Sommerfeld conditions, in which the form $pdq$ is replaced by $dp\Lambda dq$ and the vacuum energy $h/2$ by $h\nu/2$, where $\nu$ is the index of two-dimensional noncontractable cycles in $\mathfrak X$ . A study is made of smooth manifolds $\mathfrak X$ on which the index $\nu$ is integral and manifolds with conical singularities, on which $\nu$ can take half-integral values. Smooth functions $f$ on $\mathfrak X$ are associated with operators $\hat{f}$ that act on the sections of a ertain sheaf and locally have the form $\hat{f}=f(q,-ih\partial/\partial q)$, $h\to0$.